The present invention relates to the determination of a response characteristic of an nth order linear system. More particularly, the present invention relates to determination of the response characteristic from a variance measured in the output signal of the linear system.
Nth order linear systems, such as phase locked loops (PLLs) and other electrical devices that can be approximated as behaving linearly, produce an output signal in response to receiving an input signal. The output signal has a variance at any point in time from an expected or mean output signal value. The output signal and its variation over time results from the interaction of the input signal, the transfer function of the linear system, and any noise processes that are present which may include periodic processes. Noise processes may originate externally or within the linear system and this noise degrades the performance of the linear system.
When designing a linear system, the input signal, noise processes, and transfer function can be assumed for and/or estimated from simulations. However, when the linear system is to be constructed, for example, into an integrated circuit prototype, there is only access to the input and output signals. Details of the response including the noise processes and transfer function of this prototype cannot be directly measured. However, knowing the details of the response of the linear system prototype is important to confirm the design specification and simulation assumptions and to allow any design flaws and noise to be identified, fixed, and/or improved.
A conventional method of finding response characteristics for a linear system has been to apply a fast Fourier transform (FFT) approach to obtain a response characteristic such as the power spectral density (PSD) from the output signal. This approach is limited in its application because the FFT approach requires uniform sampling of the input to the FFT which is the output signal of the system under test. Uniform sampling necessitates increased recording and storage requirements and creates poor resolution for low frequencies. Furthermore, the FFT is known to be an inefficient narrowband approach when applied to broadband systems.
Therefore, there is a need for additional methods that can determine response characteristics of an nth order linear system through access to the output signal.
Embodiments of the present invention address the problems discussed above and others such as by measuring an output signal of the nth order linear system and constructing a variance record of a measurable quantity from the output signal. For example, jitter variance of a PLL may be measured as a function of time. A response characteristic of the linear system is then obtained from a mathematical relationship to the variance record. Alternatively, a PSD record may be obtained by various methods from the output signal, and the transfer function may be found by modeling an assumed response function to the PSD record to find transfer function parameters.
The response characteristic may be obtained from the variance record through numerical or analytical means or by a combination. For example, the response characteristic, such as a PSD record, may be found from a direct numerical solution to an inhomogenous Fredholm integral of the first kind, see reference below, that has been adapted to relate the variance record to an unknown transfer function. A transfer function model may also be analytically fitted to the PSD to obtain the zeros and complex and real poles.
Alternatively, a variance model that is a generic solution to the Fredholm integral may be fitted to the variance record to construct the response characteristic. By fitting the variance model to the variance record, parameters of the transfer function of the linear system, such as the damping factor and natural frequency of a second order linear system, may be found. From the transfer function parameters, the noise processes of the linear system may also be derived, such as by mathematical relationships between the transfer function and noise spectral density. The transfer function provides the zeros and complex and real poles as well as other physical parameters of the linear system such as the impulse response and/or step response.
Furthermore, by fitting a model mathematically related to the response function to the variance record or PSD record, an error estimate for the response characteristic being found may be generated. The fitting process provides a residue that results from the imperfections of the fit of the model. The total residue of imperfections can be used for relative comparisons to other attempts to attain the response characteristic, and thereby provide guidance as to accuracy.